Brady released an outline of "tax reform 2.0" before the August recess and said he plans to release legislative text early next week. The outline of the package had three components: making the individual tax cuts in the 2017 tax law permanent, providing incentives for taxpayers to save money, and providing incentives for business innovation.

"It's full steam ahead on 2.0 because the main question here is, will we make tax cuts for families and small businesses permanent as we did for corporations? The answer is yes," Brady said. "Plus it's incredibly pro-growth."

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The tax changes for individuals in the 2017 law are currently set to expire after 2025, while the tax cuts for corporations are permanent. The individual tax cuts were made temporary because lawmakers wanted to comply with budget rules that allowed the measure to pass the Senate without any support from Democrats.

The House is scheduled to be in recess for much of October so that lawmakers can campaign for the midterm elections, and lawmakers will also have a number of other issues to tackle in September.

When asked if he thinks all three parts of his package will get a floor vote before the recess, Brady said he's working to have legislation ready for a House floor vote this month.

Nearly every House Republican who voted against last year's tax law did so because they were concerned that the measure's $10,000 cap on the state and local tax (SALT) deduction would hurt their constituents. Many of the areas where the SALT deduction is popular are held by GOP House members who are in competitive reelection races, and those lawmakers may not support making the deduction cap permanent.

Brady said he still wants tax reform 2.0 to include a permanent extension of the SALT cap.